A186448 E.g.f. A(x) satisfies 2*A(x) = x*(1 + A(x) + exp(A(x))).

1, 2, 10, 88, 1144, 19856, 432464, 11348352, 348715392, 12286859008, 488470565632, 21633197775872, 1056315874429952, 56382210082129920, 3266205054434912256, 204097766901573320704, 13684668496370094407680

Offset

1,2

Comments

A(x) is e.g.f. for a(n)/(2^(n-1)/n). - Vaclav Kotesovec, Jan 26 2014

Links

Table of n, a(n) for n=1..17.

Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Formula

a(n+1) = (n!/(2*n+2)) * (Sum_{m=1..n} binomial(n+1, m) * (Sum_{i=0..n} (m^i/i!) * binomial(n-m+1, n-i))) + (n+1)^(n-1)/2 + n!/2.

a(n) ~ n^(n-2) * (1+c)^(n+1) / (2 * c^n * exp(n)), where c = LambertW(exp(-1)) = 0.278464542761... (see A202357). - Vaclav Kotesovec, Jan 26 2014

Mathematica

Table[(Rest[CoefficientList[InverseSeries[Series[(2*x)/(1+E^x+x), {x, 0, 20}], x], x]*Range[0, 20]!])[[n]] * (2^(n-1)/n), {n, 1, 20}] (* Vaclav Kotesovec, Jan 26 2014 *)
Table[n!/(2*n+2)*Sum[Binomial[n+1, m]*Sum[m^(i)/i!*Binomial[n-m+1, n-i], {i, 0, n}], {m, 1, n}]+(n+1)^(n-1)/2+n!/2, {n, 0, 20}] (* Vaclav Kotesovec after Vladimir Kruchinin, Jan 26 2014 *)

Prog

(Maxima)
a(n):=n!/(2*n+2)*sum(binomial(n+1, m)*sum(m^(i)/i!*binomial(n-m+1, n-i), i, 0, n), m, 1, n)+(n+1)^(n-1)/2+(n)!/2;

Crossrefs

Cf. A202357.

Sequence in context: A354240 A306404 A111811 * A144002 A355098 A209884

Adjacent sequences: A186445 A186446 A186447 * A186449 A186450 A186451

Keyword

nonn

Author

Vladimir Kruchinin, Feb 22 2011

Status

approved